SAT Question 49800634: Linear Equations In Two Variables answer and explanation with Desmos solution

Choice D is correct. The linear relationship between $x$ and $y$ can be represented by the equation $y=mx+b$, where $m$ is the slope of the graph of this equation in the xy-plane and $b$ is the y-coordinate of the y-intercept. The slope of a line between any two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ on the line can be calculated using the slope formula $m=\frac{y_2-y_1}{x_2-x_1}$. Based on the table, the graph contains the points $\left(-18,-48\right)$ and $\left(7,52\right)$. Substituting $\left(-18,-48\right)$ and $\left(7,52\right)$ for $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, respectively, in the slope formula yields $m=\frac{52-\left(-48\right)}{7-\left(-18\right)}$, which is equivalent to $m=\frac{100}{25}$, or $m=4$. Substituting $4$ for $m$, $-18$ for $x$, and $-48$ for $y$ in the equation $y=mx+b$ yields $-48=4\left(-18\right)+b$, or $-48=-72+b$. Adding $72$ to both sides of this equation yields $24=b$. Therefore, $m=4$ and $b=24$. Substituting $4$ for $m$ and $24$ for $b$ in the equation $y=mx+b$ yields $y=4x+24$. Thus, the equation $y=4x+24$ represents the linear relationship between $x$ and $y$. It's also given that the graph of the linear equation representing this relationship in the xy-plane passes through the point $\left(\frac{1}{7},a\right)$. Substituting $\frac{1}{7}$ for $x$ and $a$ for $y$ in the equation $y=4x+24$ yields $a=4\left(\frac{1}{7}\right)+24$, which is equivalent to $a=\frac{4}{7}+\frac{168}{7}$, or $a=\frac{172}{7}$.